Many seemingly basic questions of area, length and tangents could not be solved by the ancient geometers:
Even when solutions could be found, they were always ad hoc. As with the circle, Archimedes was also able to find the area enclosed by a line and a parabola. But his solution required an ingenious setup that was entirely different from his solution for the circle. Every property of every curve needed its own distinct solution.
This was the problem that 17th century mathematicians were aiming to address. Before calculus had been invented, René Descartes described in a 1619 letter his ambitious goal to create an “entirely new science, by which all problems that can be posed, concerning any kind of quantity, continuous or discrete, can be generally solved.”
Newton’s mentor Isaac Barrow demonstrated the fundamental theorem of calculus and came close to developing a true general calculus. In describing his unified approach to analyzing curves, he wrote: “I have previously proved a number of general properties of curves of continuous curvature, deducing them from a certain mode of construction common to all.”1
The first publication on (differential) calculus proper is credited to Gottfried Leibniz in 1684; it is from here the name “calculus” originates. To understand how Leibniz perceived his invention and its importance, we need only read the title of the paper: “A new method for maxima and minima, and for tangents, that is not hindered by fractional or irrational quantities, and a singular kind of calculus for the above mentioned.” In other words, it was one method (“singular kind of calculus”) from which multiple properties (maxima, minima, and tangents) could be obtained, applicable even to complicated curves (“not hindered by…”).
The key was clearly its general-purpose, algorithmic nature. Leibniz gave for the first time the world a tool that made the calculation of these properties easy, systematic and general. In the paper he provided not only algorithms for determining tangents, minima and maxima for general curves, but also a corresponding notation and general rules of differentiation (e.g., the addition rule, product rule, quotient rule and power rule). A few years later, Leibniz also published papers on integral calculus and on the fundamental theorem of calculus.
Independently and before Leibniz, Newton also invented a calculus he called the Method of Fluxions, though it remained unpublished until after his death. The two methods differed in important ways—certainly in terminology and symbols used—and yet generality was also key to the Method of Fluxions.